Journal of the
Korean Mathematical Society JKMS

In this paper, we will investigate the concept of the torsion-graph of an $R$-module $M$, in which the set $T(M)^*$ makes up the vertices of the corresponding torsion graph, $\Gamma(M)$, with any two distinct vertices forming an edge if $[x:M][y:M]M=0$. We prove that, if $\Gamma(M)$ contains a cycle, then $gr(\Gamma(M))\leq 4$ and $\Gamma(M)$ has a connected induced subgraph $\bar{\Gamma}(M)$ with vertex set $\{m\in T(M)^*~|~ {\rm Ann}(m)M\neq 0\}$ and diam$(\bar{\Gamma}(M) )\leq 3$. Moreover, if $M$ is a multiplication $R$-module, then $\bar{\Gamma}(M)$ is a maximal connected subgraph of $\Gamma(M)$. Also $\bar{\Gamma}(M)$ and $\bar{\Gamma}(S^{-1}M)$ are isomorphic graphs, where $S=R\setminus Z(M)$. Furthermore, we show that, if $\bar{\Gamma}(M)$ is uniquely complemented, then $S^{-1}M$ is a von Neumann regular module or $\bar{\Gamma}(M)$ is a star graph.

Keywords: torsion graph, multiplication modules, von Neumann regular modules

MSC numbers: 13A99, 05C99, 13C99